Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes

Gerd Kunert

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 6, page 1079-1109
  • ISSN: 0764-583X

Abstract

top
Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction–diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.

How to cite

top

Kunert, Gerd. "Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.6 (2001): 1079-1109. <http://eudml.org/doc/194087>.

@article{Kunert2001,
abstract = {Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction–diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.},
author = {Kunert, Gerd},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {error estimator; anisotropic solution; stretched elements; reaction diffusion equation; singularly perturbed problem; singular perturbation; error bounds; numerical example},
language = {eng},
number = {6},
pages = {1079-1109},
publisher = {EDP-Sciences},
title = {Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes},
url = {http://eudml.org/doc/194087},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Kunert, Gerd
TI - Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 6
SP - 1079
EP - 1109
AB - Singularly perturbed problems often yield solutions with strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction–diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.
LA - eng
KW - error estimator; anisotropic solution; stretched elements; reaction diffusion equation; singularly perturbed problem; singular perturbation; error bounds; numerical example
UR - http://eudml.org/doc/194087
ER -

References

top
  1. [1] M. Ainsworth and I. Babuška, Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. 36 (1999) 331–353. Zbl0948.65114
  2. [2] M. Ainsworth and J. Oden, A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000). Zbl1008.65076MR1885308
  3. [3] L. Angermann, Balanced a-posteriori error estimates for finite volume type discretizations of convection-dominated elliptic problems. Computing 55 (1995) 305–323. Zbl0839.65114
  4. [4] T. Apel and G. Lube, Anisotropic mesh refinement in stabilized Galerkin methods. Numer. Math. 74 (1996) 261–282. Zbl0878.65097
  5. [5] T. Apel and S. Nicaise, The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges. Math. Methods Appl. Sci. 21 (1998) 519–549. Zbl0911.65107
  6. [6] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978) 736–754. Zbl0398.65069
  7. [7] N.S. Bakhvalov, Optimization of methods for the solution of boundary value problems in the presence of a boundary layer. Zh. Vychisl. Mat. i Mat. Fiz. 9 (1969) 841–859. In Russian. Zbl0208.19103
  8. [8] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 283–301. Zbl0569.65079
  9. [9] M. Beckers, Numerical Integration in High Dimensions. Ph.D. Thesis, Katholieke Universiteit Leuven / Louvain, Belgium (1992). 
  10. [10] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam, New York, Oxford (1978). Zbl0383.65058MR520174
  11. [11] M. Dobrowolski, S. Gräf and C. Pflaum, On a posteriori error estimators in the finite element method on anisotropic meshes. ETNA, Electron. Trans. Numer. Anal. 8 (1999) 36–45. Zbl0934.65122
  12. [12] P. Keast, Moderate-degree tetrahedral quadrature formulas. Comput. Methods Appl. Mech. Engrg. 55 (1986) 339–348. Zbl0572.65008
  13. [13] G. Kunert, A Posteriori Error Estimation for Anisotropic Tetrahedral and Triangular Finite Element Meshes. Logos Verlag, Berlin (1999). Also Ph.D. Thesis, TU Chemnitz, http://archiv.tu-chemnitz.de/pub/1999/0012/index.html Zbl0919.65066
  14. [14] G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86 (2000) 471–490. DOI 10.1007/s002110000170. Zbl0965.65125
  15. [15] G. Kunert, Towards anisotropic mesh construction and error estimation in the finite element method. To appear in Numer. Meth. Partial Differential Equations. Preprint SFB393/0001, TU Chemnitz (2000). Also http://archiv.tu-chemnitz.de/pub/2000/0066/index.html Zbl1041.65097MR1919601
  16. [16] G. Kunert, A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39 (2001) 668–689. Zbl1004.65112
  17. [17] G. Kunert, A note on the energy norm for a singularly perturbed model problem. Preprint SFB393/01-02, TU Chemnitz (2001). Also http://archiv.tu-chemnitz.de/pub/2001/0006/index.html Zbl1041.65072MR1954563
  18. [18] G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. To appear in Adv. Comp. Math. Zbl1049.65121MR1887735
  19. [19] G. Kunert and R. Verfürth, Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86 (2000) 283–303. DOI 10.1007/s002110000152. Zbl0964.65120
  20. [20] J. Peraire, M. Vahdati, K. Morgan and O.C. Zienkiewicz, Adaptive remeshing for compressible flow computation. J. Comput. Phys. 72 (1987) 449–466. Zbl0631.76085
  21. [21] W. Rick, H. Greza and W. Koschel, FCT-solution on adapted unstructured meshes for compressible high speed flow computations. in Flow Simulation with High-Performance Computers I, in Notes Numer. Fluid Mech. 38, E.H. Hirschel, Ed., Vieweg (1993) 334–438 . 
  22. [22] H.-G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer, Berlin (1996). Zbl0844.65075MR1477665
  23. [23] K.G. Siebert, An a posteriori error estimator for anisotropic refinement. Numer. Math. 73 (1996) 373–398. Zbl0873.65098
  24. [24] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50 (1994) 67–83. Zbl0811.65089
  25. [25] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Chichester, Stuttgart (1996). Zbl0853.65108
  26. [26] R. Verfürth, Robust a posteriori error estimators for singularly perturbed reaction–diffusion equations. Numer. Math. 78 (1998) 479–493. Zbl0887.65108
  27. [27] R. Vilsmeier and D. Hänel, Computational aspects of flow simulation in three dimensional, unstructured, adaptive grids, in Flow Simulation with High-Performance Computers II, in Notes Numer. Fluid Mech. 52, E.H. Hirschel, Ed., Vieweg (1996) 431–44. Zbl0876.76065
  28. [28] O.C. Zienkiewicz and J. Wu, Automatic directional refinement in adaptive analysis of compressible flows. Internat. J. Numer. Methods Engrg. 37 (1994) 2189–2210 . Zbl0810.76045

NotesEmbed ?

top

You must be logged in to post comments.